Three-dimensional mesh generating method, magnetic field analysis method for rotating machine, three-dimensional mesh generating device, magnetic field analysis device for rotating machine, computer program, and recording medium

ABSTRACT

A two-dimensional mesh is generated on a plane perpendicular to the rotation axis. At this time, a ring-shaped gap is provided between the rotor and the stator, and portions facing the ring-shaped gap are equally divided into the same number of parts. An initial three-dimensional mesh is generated by joining together a plurality of two-dimensional meshes in the direction of the rotation axis while rotating the two-dimensional meshes. A boundary surface is formed in a cylindrical gap composed of a stack of the ring-shaped gaps, and a three-dimensional mesh is generated by filling the cylindrical gap with a plurality of polyhedrons, including polyhedrons comprising each of surface elements constituting the stator-side mesh surface, rotor-side mesh surface and boundary surface as one face.

This application is the national phase under 35 U.S.C. § 371 of PCTInternational Application No. PCT/JP02/09131 which has an Internationalfiling date of Sep. 6, 2002, which designated the United States ofAmerica.

TECHNICAL FIELD

The present invention relates to: a method for generating athree-dimensional mesh representing a rotating machine having skew,including a spatial area, for the analysis of electromagnetic fieldusing a finite element method; a three-dimensional mesh generatingapparatus for use in implementing the method; a computer program forrealizing a computer as the three-dimensional mesh generating apparatus;a memory product readable by a computer storing the computer program; amethod for analyzing the magnetic field of a rotating machine by using athree-dimensional mesh; a magnetic field analyzing apparatus for use inimplementing the method; a computer program for realizing a computer asthe magnetic field analyzing apparatus; and a memory product readable bya computer storing the computer program.

BACKGROUND ART

Rotating machines are important parts that are incorporated and used invarious apparatuses, and improvements are repeatedly made for highperformance. In order to support the designing of a new rotatingmachine, numerical analysis of the performance by a finite elementmethod is generally performed by changing the shape of the rotatingmachine.

The finite element method is a method for performing numericalcalculation by representing an object to be analyzed by a combination ofa plurality of polyhedral elements, and is widely used for the analysisof the structure of three-dimensional objects. In the case where thefinite element method is used for the numerical analysis of a rotatingmachine, in order to analyze the magnetic field between a stator and arotor, it is necessary to generate a three-dimensional mesh representingthe spatial area between the stator and the rotor as well as the statorand the rotor by a combination of a plurality of polyhedral elements.Moreover, in order to analyze the magnetic field while rotating therotor, it is necessary to take into account the rotational motion ingenerating the three-dimensional mesh.

Conventionally, a three-dimensional mesh of the rotating machine isgenerated as follows. A boundary surface in the form of a cylindricalsurface is set in the spatial area of the rotating machine, and a spaceon the stator side and a space on the rotor side including the spatialarea with the boundary surface therebetween are represented bycombinations of a plurality of polyhedrons, respectively, on atwo-dimensional plane perpendicular to the rotation axis to generatetwo-dimensional meshes. At this time, portions of the two-dimensionalmeshes on the stator side and the rotor side, which come into contactwith the boundary surface, are equally divided in the rotation directionand arranged to match each other at the boundary surface. Next, thegenerated two-dimensional meshes are stacked in the direction of therotation axis to generate a three-dimensional mesh representing therotating machine including the spatial area. By shifting thethree-dimensional mesh on the rotor side from the boundary surface byone element with respect to the stator side, the rotor can be rotated,and the magnetic field of the rotating machine can be analyzed whilerotating the rotor.

Further, Japanese Patent Application Laid-Open No. 2001-155055 disclosesa method for generating a three-dimensional mesh of a rotating machineby generating three-dimensional meshes on the stator side and the rotorside independently of each other, providing a gap between them, andgenerating a three-dimensional mesh of the gap by filling the gap withpolyhedrons by an automatic element division method. In the case where athree-dimensional mesh generated by this method is used, analysis of themagnetic field of the rotating machine while rotating the rotor can beperformed by regenerating a three-dimensional mesh of the gap wheneverthe rotor is rotated.

One of the problems of the rotating machine is the torque variation,which occurs because the magnitude of magnetic flux that generates therotating force in the rotor changes depending on the positionalrelationship between the stator and the rotor, and this torque variationwill cause vibration and noise of the rotating machine. In order toreduce the torque variation, used is a rotating machine having skew thatis a structure of the stator or the rotor twisted in the direction ofthe rotation axis. However, in order to perform numerical analysis ofthe rotating machine having skew by the finite element method, there isthe problem that it is impossible to use a conventionalthree-dimensional mesh generating method in which two-dimensional meshesgenerated on a plane perpendicular to the rotation axis are stacked inthe direction of the rotation axis, because the shape of the rotatingmachine is not symmetric about the plane perpendicular to the rotationaxis.

In addition, according to the method disclosed in Japanese PatentApplication Laid-Open No. 2001-155055, since the three-dimensionalmeshes on the stator side and the rotor side are generated independentlyof each other, it is possible to generate a three-dimensional mesh of arotating machine having skew, but this method has the problem that ittakes a long time for calculation in the analysis because athree-dimensional mesh needs to be regenerated whenever rotation isperformed, and the problem that a process for giving thethree-dimensional mesh periodicity in the rotation direction isadditionally required because the three-dimensional mesh regeneratedwhenever rotation is performed is irregular.

DISCLOSURE OF THE INVENTION

The present invention has been made with the aim of solving the aboveproblems, and it is an object of the present invention to provide athree-dimensional mesh generating method capable of generating athree-dimensional mesh of a rotating machine having skew by providing agap between a rotor and a stator, equally dividing both sides of the gapto generate two-dimensional meshes on the stator side and the rotor sideon a two-dimensional plane perpendicular to the rotation axis, stackingthe two-dimensional meshes while twisting them to generate an initialthree-dimensional mesh having skew, forming a boundary surface betweenthe stator side and the rotor side of the initial three-dimensional meshby projecting a stator-side or rotor-side surface mesh, and filling thespaces between the boundary surface and the stator-side and rotor-sidesurface meshes with a plurality of polyhedrons including respectivepolyhedrons constituting the boundary surface and the respective surfacemeshes, and to provide a three-dimensional mesh generating apparatus foruse in implementing the method, a computer program for realizing acomputer as the three-dimensional mesh generating apparatus, a memoryproduct readable by a computer storing the computer program, a methodfor analyzing the magnetic field of a rotating machine in a shortcalculation time by rotating the three-dimensional mesh on the rotorside from the boundary surface, a magnetic field analyzing apparatus foruse in implementing the method, a computer program for realizing acomputer as the magnetic field analyzing apparatus, and a memory productreadable by a computer storing the computer program.

Another object of the present invention is to provide athree-dimensional mesh generating method capable of giving athree-dimensional mesh periodicity in the rotation direction, withoutrequiring an additional process, by generating the three-dimensionalmesh by associating each of surface elements constituting the surfacemeshes on the stator side and the rotor side with surface elementsconstituting the boundary surface, and filling the space betweencorresponding surface elements with polyhedrons, and to provide athree-dimensional mesh generating apparatus for use in implementing themethod, a computer program for realizing a computer as thethree-dimensional mesh generating apparatus, and a memory productreadable by a computer storing the computer program.

A three-dimensional mesh generating method according to the firstinvention is a method for generating a three-dimensional meshrepresenting a rotating machine with a stator or a rotor having atwisted structure in a direction of a rotation axis of the rotor,including a spatial area between the stator and the rotor, by acombination of a plurality of polyhedrons, and characterized by:generating a two-dimensional mesh in which a ring-shaped gap is providedaround the rotation axis in the spatial area, portions facing each otherwith the ring-shaped gap therebetween are equally divided into mutuallyequal number of parts, and a stator-side portion and a rotor-sideportion, excluding the ring-shaped gap, are represented by a combinationof a plurality of polyhedrons on a plane perpendicular to the rotationaxis; generating an initial three-dimensional mesh by joining together aplurality of the two-dimensional meshes with the stator-side portion andthe rotor-side portion relatively rotated on the rotation axis accordingto the twisted structure, in a direction of the rotation axis accordingto a same rule in the stator-side portion and the rotor-side portion;forming a boundary surface constructed by a mesh surface obtained byconcentrically projecting, into a cylindrical gap composed of a stack ofthe ring-shaped gaps, any one of a stator-side mesh surface and arotor-side mesh surface which face each other with the cylindrical gaptherebetween; and filling spaces between the boundary surface and thestator-side mesh surface and rotor-side mesh surface with a plurality ofpolyhedrons, including polyhedrons comprising each of surface elementsconstituting the boundary surface, the stator-side mesh surface and therotor-side mesh surface as one face, to generate the three-dimensionalmesh.

A three-dimensional mesh generating method according to the secondinvention is characterized in that the two-dimensional mesh is composedof a combination of a plurality of quadrangles, and the initialthree-dimensional mesh is generated by joining together a plurality ofthe two-dimensional meshes in the direction of the rotation axis so thatcorresponding nodes in the two-dimensional meshes are connected by astraight line.

A three-dimensional mesh generating method according to the thirdinvention is characterized by dividing each of quadrangular elementsconstituting the boundary surface into two triangular elements arrangedin a direction in which the boundary surface is twisted with respect tothe stator-side mesh surface or the rotor-side mesh surface; connectingeach of nodes constituting the stator-side mesh surface and therotor-side mesh surface to a closest node among a plurality of nodesconstituting the boundary surface by a straight line; and filling aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base.

A three-dimensional mesh generating method according to the fourthinvention is a method for generating a three-dimensional meshrepresenting a rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons by acomputer, and characterized by comprising steps of receiving from aninput unit and storing into a storage unit an initial three-dimensionalmesh in which a cylindrical gap is provided in the spatial area betweenthe stator and the rotor of the rotating machine, portions facing eachother with the cylindrical gap therebetween are divided into mutuallyequal lengths perpendicular to a rotation axis of the rotating machinein a direction of the rotation axis and equally divided into mutuallyequal number of parts in a direction around the rotation axis, and astator-side portion and a rotor-side portion of the rotating machine,excluding the cylindrical gap, are represented by a combination of aplurality of polyhedrons; forming a boundary surface by a mesh surfaceobtained by concentrically projecting, into the cylindrical gap, any oneof a stator-side mesh surface and a rotor-side mesh surface which faceeach other with the cylindrical gap therebetween, and storing theboundary surface into the storage unit; dividing each of quadrangularelements constituting the boundary surface into two triangular elementsarranged in a direction tilted with respect to each of quadrangularelements constituting the stator-side mesh surface or the rotor-sidemesh surface, and storing them into the storage unit; connecting each ofnodes constituting the stator-side mesh surface and the rotor-side meshsurface to a closest node among a plurality of nodes constituting theboundary surface by a straight line, and storing them into the storageunit; and filling a space between each of surface elements constitutingthe stator-side mesh surface and rotor-side mesh surface and acombination of the two triangular elements connected to the surfaceelement by straight lines, with four tetrahedrons, including twotetrahedrons comprising each of the two triangular elements as one face,and one quadrangular pyramid comprising the surface element as a base,and storing them into the storage unit.

A magnetic field analyzing method for a rotating machine according tothe fifth invention is a method for analyzing the magnetic field of arotating machine by a finite element method using a three-dimensionalmesh representing the rotating machine, including a spatial area betweena stator and a rotor, by a combination of a plurality of polyhedrons,and characterized by generating a three-dimensional mesh representing arotating machine to be analyzed, by using the three-dimensional meshgenerating method of any one of the first through fourth inventions,rotating a rotor side of the three-dimensional mesh by shifting theelements from the boundary surface, and analyzing the magnetic field bythe finite element method.

A three-dimensional mesh generating apparatus according to the sixthinvention is an apparatus for generating a three-dimensional meshrepresenting a rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons, andcharacterized by comprising: means for receiving an initialthree-dimensional mesh in which a cylindrical gap is provided in thespatial area between the stator and the rotor of the rotating machine,portions facing each other with the cylindrical gap therebetween aredivided into mutually equal lengths perpendicular to a rotation axis ofthe rotating machine in a direction of the rotation axis and equallydivided into mutually equal number of parts in a direction around therotation axis, and a stator-side portion and a rotor-side portion of therotating machine, excluding the cylindrical gap, are represented by acombination of a plurality of polyhedrons; means for forming a boundarysurface by a mesh surface obtained by concentrically projecting, intothe cylindrical gap, any one of a stator-side mesh surface and arotor-side mesh surface which face each other with the cylindrical gaptherebetween; means for dividing each of quadrangular elementsconstituting the boundary surface into two triangular elements arrangedin a direction in which the boundary surface is tilted with respect tothe stator-side mesh surface or the rotor-side mesh surface; means forconnecting each of nodes constituting the stator-side mesh surface andthe rotor-side mesh surface to a closest node among a plurality of nodesconstituting the boundary surface by a straight line; and means forfilling a space between each of surface elements constituting thestator-side mesh surface and rotor-side mesh surface and a combinationof the two triangular elements connected to the surface element bystraight lines, with four tetrahedrons, including two tetrahedronscomprising each of the two triangular elements as one face, and onequadrangular pyramid comprising the surface element as a base.

A magnetic field analyzing apparatus for a rotating machine according tothe seventh invention is an apparatus for analyzing a magnetic field ofa rotating machine by a finite element method using a three-dimensionalmesh representing the rotating machine, including a spatial area betweena stator and a rotor, by a combination of a plurality of polyhedrons,and characterized by comprising: means for generating athree-dimensional mesh representing a rotating machine to be analyzed,by using the three-dimensional mesh generating apparatus of the sixthinvention; and means for rotating a rotor side of the three-dimensionalmesh by shifting the elements from the boundary surface, and analyzingthe magnetic field by the finite element method.

A computer program according to the eighth invention is a computerprogram for causing a computer to generate a three-dimensional meshrepresenting a rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons, byusing an initial three-dimensional mesh in which a cylindrical gap isprovided in the spatial area between the stator and the rotor of therotating machine, portions facing each other with the cylindrical gaptherebetween are divided into mutually equal lengths perpendicular to arotation axis of the rotating machine in a direction of the rotationaxis and equally divided into mutually equal number of parts in adirection around the rotation axis, and a stator-side portion and arotor-side portion of the rotating machine, excluding the cylindricalgap, are represented by a combination of a plurality of polyhedrons, andcharacterized by comprising steps of causing the computer to form aboundary surface by a mesh surface obtained by concentricallyprojecting, into the cylindrical gap, any one of a stator-side meshsurface and a rotor-side mesh surface which face each other with thecylindrical gap therebetween; causing the computer to divide each ofquadrangular elements constituting the boundary surface into twotriangular elements arranged in a direction in which the boundarysurface is tilted with respect to the stator-side mesh surface or therotor-side mesh surface; causing the computer to connect each of nodesconstituting the stator-side mesh surface and the rotor-side meshsurface to a closest node among a plurality of nodes constituting theboundary surface by a straight line; and causing the computer to fill aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base.

A computer program according to the ninth invention is a computerprogram for causing a computer to analyze a magnetic field of a rotatingmachine by a finite element method using a three-dimensional meshrepresenting a rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons, andcharacterized by comprising steps of: causing the computer to generate athree-dimensional mesh representing a rotating machine to be analyzed,by using the computer program of the eighth invention; and causing thecomputer to rotate a rotor side of the three-dimensional mesh byshifting the elements from the boundary surface and analyze the magneticfield by the finite element method.

A computer-readable memory product according to the tenth invention is amemory product readable by a computer storing a computer program forcausing a computer to generate a three-dimensional mesh representing arotating machine, including a spatial area between a stator and a rotor,by a combination of a plurality of polyhedrons, by using an initialthree-dimensional mesh in which a cylindrical gap is provided in thespatial area between the stator and the rotor of the rotating machine,portions facing each other with the cylindrical gap therebetween aredivided into mutually equal lengths perpendicular to a rotation axis ofthe rotating machine in a direction of the rotation axis and equallydivided into mutually equal number of parts in a direction around therotation axis, and a stator-side portion and a rotor-side portion of therotating machine, excluding the cylindrical gap, are represented by acombination of a plurality of polyhedrons, and characterized by storinga computer program comprising steps of: causing the computer to form aboundary surface by a mesh surface obtained by concentricallyprojecting, into the cylindrical gap, any one of a stator-side meshsurface and a rotor-side mesh surface which face each other with thecylindrical gap therebetween; causing the computer to divide each ofquadrangular elements constituting the boundary surface into twotriangular elements arranged in a direction in which the boundarysurface is tilted with respect to the stator-side mesh surface or therotor-side mesh surface; causing the computer to connect each of nodesconstituting the stator-side mesh surface and the rotor-side meshsurface to a closest node among a plurality of nodes constituting theboundary surface by a straight line; and causing the computer to fill aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base.

A computer-readable memory product according to the eleventh inventionis a memory product readable by a computer storing a computer programfor causing a computer to analyze a magnetic field of a rotating machineby a finite element method using a three-dimensional mesh representing arotating machine, including a spatial area between a stator and a rotor,by a combination of a plurality of polyhedrons, and characterized bystoring a computer program comprising steps of causing the computer togenerate a three-dimensional mesh representing a rotating machine to beanalyzed, by using the computer program of the tenth invention; andcausing the computer to rotate a rotor side of the three-dimensionalmesh by shifting the elements from the boundary surface and analyze themagnetic field by the finite element method.

FIG. 1 is an explanatory view showing the procedure of athree-dimensional mesh generating method of the first invention. In thefirst invention, a ring-shaped gap G1 is provided between the rotor andthe stator on a two-dimensional plane perpendicular to the rotation axisas shown in FIG. 1(a), and both sides of the ring-shaped gap G1 areequally divided to generate a two-dimensional mesh of the stator sideand the rotor side as shown in FIG. 1(b). Next, as shown in FIG. 1(c),an initial three-dimensional mesh having skew is generated by joiningtogether the two-dimensional meshes with the stator side and the rotorside relatively rotated according to the skew structure, in thedirection of the rotation axis. Next, as shown in FIG. 1(d), a boundarysurface SL is formed by projecting, into the cylindrical gap, any one ofa stator-side mesh surface ST and a rotor-side mesh surface RT whichface each other with a cylindrical gap G2 therebetween. Next, athree-dimensional mesh is generated as shown in FIG. 1(e) by filing thecylindrical gap G2 with a plurality of polyhedrons including polyhedronscomprising each of surface elements constituting the boundary surfaceSL, the stator-side mesh surface ST and the rotor-side mesh surface RTas one face. Since portions of the stator side and the rotor side of thethree-dimensional mesh which come into contact with each other at theboundary surface are composed of elements having mutually equal size inthe rotation direction, it is possible to rotate the rotor side of athree-dimensional mesh representing a rotating machine having skew byshifting the elements from the boundary surface.

In the second invention, the two-dimensional mesh is composed ofquadrangles, an initial three-dimensional mesh is generated byconnecting corresponding nodes in the two-dimensional meshes, and thuseach element of the initial three-dimensional mesh is a hexahedron. Inthe infinite element method, since the calculation accuracy is improvedby using hexahedral elements than by using tetrahedral elements, it ispossible to generate a three-dimensional mesh with high calculationaccuracy by this method.

In the third invention, each of the surface elements constituting thestator-side mesh surface ST and the rotor-side mesh surface RT isassociated with surface elements constituting the boundary surface SL,and the space between corresponding surface elements is filled with onequadrangular pyramid and four tetrahedrons. Consequently, thethree-dimensional mesh is made periodic in the rotation directionwithout requiring an additional process.

In the fourth, sixth, eighth and tenth inventions, input of an initialthree-dimensional mesh having a cylindrical gap between the stator-sideportion and the rotor-side portion is received, a boundary surface SL isformed by projecting, into the cylindrical gap G2, any one of astator-side mesh surface ST and a rotor-side mesh surface RT which faceeach other with the cylindrical gap G2 therebetween, each of surfaceelements constituting the stator-side mesh surface ST and the rotor-sidemesh surface RT is associated with surface elements constituting theboundary surface SL, and the space between corresponding surfaceelements is filled with polyhedrons including one quadrangular pyramidand four tetrahedrons. Consequently, even for a three-dimensional meshrepresenting a rotating machine having skew, it is possible to generatea three-dimensional mesh that allows rotation of the rotor side byshifting the elements from the boundary surface, and has periodicity inthe rotation direction, in high calculation accuracy.

In the fifth, seventh, ninth, and eleventh inventions, since themagnetic field of the rotating machine is analyzed by the finite elementmethod using the generated three-dimensional mesh, it is possible toperform accurate magnetic field analysis with shorter calculation time.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an explanatory view showing the procedure of athree-dimensional mesh generating method according to the firstinvention;

FIG. 2 is a block diagram showing a three-dimensional mesh generatingapparatus according to the present invention;

FIG. 3 is a partially cut perspective view showing an example of thestructure of a rotating machine having skew;

FIG. 4 is a schematic view showing an example of a two-dimensional mesh;

FIG. 5 is a schematic view for explaining the process of generating aninitial three-dimensional mesh;

FIG. 6 is a perspective view showing a part of the initialthree-dimensional mesh;

FIG. 7 is a flowchart for explaining the flow of processes performed bythe three-dimensional mesh generating apparatus;

FIG. 8 is a flowchart for explaining the procedure of a sub-routine forgenerating a boundary surface;

FIG. 9 is a schematic view showing the boundary surface;

FIG. 10 is a flowchart for explaining the procedure of a sub-routine forcompleting a three-dimensional mesh;

FIG. 11 is a schematic view for explaining an example of correspondencebetween surface elements;

FIG. 12 is a perspective view showing examples of quadrangular elementsand tetrahedral elements to be generated;

FIG. 13 is a perspective view showing a three-dimensional mesh between astator-side mesh surface and the boundary surface;

FIG. 14 is a perspective view showing a three-dimensional mesh between arotor-side mesh surface and the boundary surface;

FIG. 15 is a perspective view showing a part of a completedthree-dimensional mesh;

FIG. 16 is a block diagram showing a magnetic field analyzing apparatusfor a rotating machine according to the present invention;

FIG. 17 is a flowchart showing the flow of processes performed by themagnetic field analyzing apparatus for a rotating machine according tothe present invention; and

FIG. 18 is a perspective view showing a part of three-dimensional meshrotated.

BEST MODE FOR CARRYING OUT THE INVENTION

The following description will specifically explain the presentinvention with reference to the drawings illustrating an embodimentthereof.

FIG. 2 is a block diagram showing a three-dimensional mesh generatingapparatus according to the present invention. In the figure, 1represents a three-dimensional mesh generating apparatus of the presentinvention implemented using a computer, which comprises: a CPU 11 forperforming operations; a RAM 12; an external memory device 13 such as aCD-ROM drive; and an internal memory device 14 such as a hard disk,reads a computer program 20 of the present invention from a memoryproduct 2 such as a CD-ROM of the present invention by the externalmemory device 13, stores the read computer program 20 into the internalmemory device 14, and loads the computer program 20 into the RAM 12, andthe CPU 11 executes processes necessary for the three-dimensional meshgenerating apparatus 1, based on the computer program 20. Thethree-dimensional mesh generating apparatus 1 comprises an input device15 such as a keyboard or a mouse, and an output device 16 such as aliquid crystal display or a CRT display, and receives operations, suchas input of data, from an operator.

Moreover, the three-dimensional mesh generating apparatus 1 comprises acommunication interface 17, and may be arranged to download the computerprogram 20 of the present invention from a server device 3 connected tothe communication interface 17 and execute the processes by the CPU 11.

FIG. 3 is a partially cut perspective view showing an example of thestructure of a rotating machine having skew. A rotor having a permanentmagnet is positioned in the center of the rotating machine, a statorusing an electromagnet is positioned around the rotor, and the rotor hasskew. The change in the magnetic field with respect to rotation whichaffects the whole rotor is reduced by the skew, and the torque variationof the rotating machine becomes smaller.

The following description will explain a three-dimensional meshgenerating method of the present invention by illustrating thegeneration of a three-dimensional mesh representing the rotating machineshown in FIG. 3 as an example.

First, a two-dimensional mesh is generated by an operation of anoperator using a CAD, etc. FIG. 4 is a schematic view showing an exampleof the two-dimensional mesh, and one fourth of the entire mesh isillustrated in the figure. In response to the operation of the operator,a cross sectional shape model representing a cross section of therotating machine as shown in FIG. 4(a) is created using a CAD, etc., aring-shaped gap G1 is provided between the rotor and the stator, andportions other than the ring-shaped gap G1 are divided into a pluralityof quadrangles as shown in FIG. 4(b) to generate a two-dimensional mesh.At this time, a portion on the rotor side and a portion on the statorside which face each other with the ring-shaped gap G1 therebetween aredivided into mutually equal number of parts. Note that althoughquadrangles are mentioned as polyhedrons constituting thetwo-dimensional mesh, it may be possible to divide the portions intotriangles.

Next, in response to the operation of the operator, the two-dimensionalmeshes generated are joined together in the direction of the rotationaxis to generate an initial three-dimensional mesh. FIG. 5 is aschematic view for explaining the processes of generating the initialthree-dimensional mesh. First, one layer of two-dimensional mesh asshown in FIG. 5(a) is stacked in the direction of the rotation axis, andthe rotor side of the two-dimensional mesh is rotated according to thestructure of the skew. In the figure, M1 represents the firsttwo-dimensional mesh, and M2 represents the stacked two-dimensionalmesh. Next, corresponding nodes in the two-dimensional meshes areconnected by a straight line between the stacked two-dimensional meshesto generate one layer of initial three-dimensional mesh. Further, asshown in FIG. 5(c), the next two-dimensional mesh M3 is stacked and therotor side is rotated, and the same operation is repeated until theinitial three-dimensional mesh representing the structure of therotating machine is completed. FIG. 6 is a perspective view showing apart of the initial three-dimensional mesh. On the stator side, theinitial three-dimensional mesh is generated by stacking two-dimensionalmeshes parallel to the rotation axis, while, on the rotor side, theinitial three-dimensional mesh having a twisted structure according tothe structure of the skew is generated. A cylindrical gap G2 isgenerated between the stator side and the rotor side by a stack of thering-shaped gaps G1.

Next, a three-dimensional mesh is generated by the processes using thethree-dimensional mesh generating apparatus 1. FIG. 7 is a flowchart forexplaining the flow of processes performed by the three-dimensional meshgenerating apparatus 1. First, the three-dimensional mesh generatingapparatus 1 receives input of the initial three-dimensional meshrepresenting the structure of the rotating machine entered by theoperation of the operator (S1).

Next, the three-dimensional mesh generating apparatus 1 generates aboundary surface in the cylindrical gap G2 (S2). FIG. 8 is a flowchartfor explaining the procedure of a sub-routine for generating theboundary surface of step S2, and FIG. 9 is a schematic view showing theboundary surface. As shown in FIG. 9(a), first, the three-dimensionalmesh generating apparatus 1 generates a boundary surface SL byprojecting, into the cylindrical gap G2, a rotor-side mesh surface RT ofthe initial three-dimensional mesh on the rotor side which faces thecylindrical gap G2 (S21). At this time, it is also possible to generatethe boundary surface SL by projecting a stator-side mesh surface STfacing the rotor-side mesh surface RT with the cylindrical gap G2therebetween, instead of the rotor-side mesh surface RT, or by setting acylindrical boundary surface SL first and projecting the mesh of thestator-side mesh surface ST or the rotor-side mesh surface RT onto theset boundary surface SL.

Next, as shown in FIG. 9(b), the three-dimensional mesh generatingapparatus 1 divides each of quadrangular surface elements constitutingthe boundary surface SL into two triangular surface elements arranged ina direction in which the boundary surface SL is twisted with respect tothe stator-side mesh surface ST or the rotor-side mesh surface RT so asto produce the boundary surface SL by a combination of the triangularelements (S22), and completes the sub-routine for generating theboundary surface and returns to the main process.

Next, the three-dimensional mesh generating apparatus 1 fills thecylindrical gap G2 with a plurality of polyhedrons to complete thethree-dimensional mesh (S3). FIG. 10 is a flowchart for explaining theprocedure of a sub-routine for completing the three-dimensional mesh ofstep S3. The three-dimensional mesh generating apparatus 1 connects eachof the nodes constituting the stator-side mesh surface ST and therotor-side mesh surface RT to a node on the boundary surface SL locatedat the shortest distance by a straight line (S31), and associates eachof the quadrangular surface elements constituting the stator-side meshsurface ST and the rotor-side mesh surface RT with two triangularsurface elements constituting the boundary surface SL connected by thestraight lines (S32).

FIG. 11 is a schematic view for explaining an example of correspondencebetween surface elements, wherein a part of the stator-side mesh surfaceST and the rotor-side mesh surface RT is shown by an alternate long andtwo short dashes line, a part of the boundary surface SL is shown by asolid line, and the straight line connecting the nodes is shown by abroken line. The nodes of the rotor-side mesh surface RT and the nodesof the boundary surface SL are connected so that the nodes at theshortest distance are connected by a straight line, and quadrangularsurface elements A, B and C constituting the rotor-side mesh surface RTcorrespond respectively to combinations of triangular surface elementsa1 and a2, b1 and b2, and c1 and c2 constituting the boundary surfaceSL. Besides, the nodes of the stator-side mesh surface ST and the nodesof the boundary surface SL are connected so that the nodes at theshortest distance are connected by a straight line, and quadrangularsurface elements D, E and F constituting the stator-side mesh surface STcorrespond respectively to combinations of triangular surface elementsd1 and d2, e1 and b2, and c1 and c2 constituting the boundary surfaceSL.

Next, the three-dimensional mesh generating apparatus 1 generates aquadrangular pyramid element between corresponding one quadrangularsurface element and two triangular surface elements so that thequadrangular surface element is the base (S33), generates fourtetrahedral elements including two tetrahedrons having each of the twotriangular surface elements as one face (S34), and fills the spacebetween corresponding one quadrangular surface element and twotriangular surface elements with one quadrangular pyramid element andfour tetrahedral elements.

FIG. 12 is a perspective view showing examples of quadrangular elementsand a tetrahedral element to be generated. In the space between thesurface element A and the surface elements a1 and id shown in FIG. 11,as shown in FIG. 12(a), a quadrangular pyramid element “adheg” includinga quadrangle “adhe” that is the surface element A as the base, atetrahedral element “fgca” including a triangle “fgc” that is thesurface element a1 as one face, and a tetrahedral element “fbca”including a triangle “fbc” that is the surface element a2 as one faceare generated, and further a tetrahedral element “acdg” and atetrahedral element “aefg” are generated. In the space between thesurface element D and the surface elements d1 and d2 shown in FIG. 11,as shown in FIG. 12(b), a quadrangular pyramid element “cbfga” includinga quadrangle “cbfg” that is the surface element D as the base, atetrahedral element “hdeg” including a triangle “hde” that is thesurface element d1 as one face, and a tetrahedral element “deag”including a triangle “dea” that is the surface element a2 as one faceare generated, and further a tetrahedral element “egfa” and atetrahedral element “acdg” are generated. In the space between thesurface element E and the surface elements e1 and b2 shown in FIG. 11,as shown in FIG. 12(c), a quadrangular pyramid element “cbfge” includinga quadrangle “cbfg” that is the surface element E as the base, atetrahedral element “ahec” including a triangle “ahe” that is thesurface element e1 as one face, and a tetrahedral element “dhac”including a triangle “dha” that is the surface element b2 as one faceare generated, and further a tetrahedral element “abce” and atetrahedral element “ehgc” are generated.

By performing the filling periodically in the rotation direction, thespaces between the stator-side mesh surface ST and rotor-side meshsurface RT and the boundary surface SL are represented by combinationsof a plurality of polyhedrons, and a three-dimensional mesh representingthe rotating machine including the cylindrical gap G2 by combinations ofa plurality of polyhedrons is completed. FIG. 13 is a perspective viewshowing the three-dimensional mesh between the stator-side mesh surfaceST and the boundary surface SL, wherein FIG. 13(a) shows thethree-dimensional mesh from the boundary surface SL side, and FIG. 13(b)shows the three-dimensional mesh from the stator-side mesh surface STside. FIG. 14 is a perspective view showing the three-dimensional meshbetween the rotor-side mesh surface RT and the boundary surface SL,wherein FIG. 14(a) shows the three-dimensional mesh from the rotor-sidemesh surface RT side, and FIG. 14(b) shows the three-dimensional meshfrom the boundary surface SL side. The cylindrical gap G2 is filled witha plurality of polyhedrons periodically in the rotation direction. FIG.15 is a perspective view showing a part of the completedthree-dimensional mesh. All the portions including the cylindrical gapG2 are represented by combinations of a plurality of polyhedrons, andthe stator-side three-dimensional mesh and the rotor-sidethree-dimensional mesh match each other at the boundary surface SL.

The three-dimensional mesh generating apparatus 1 finishes thesub-routine for completing a three-dimensional mesh and returns to themain process, thereby completing the processes of generating athree-dimensional mesh.

With the use of the three-dimensional mesh generating apparatus 1 of thepresent invention, a three-dimensional mesh of a rotating machine havingskew is generated by the above-mentioned procedure. An initialthree-dimensional mesh is generated by stacking two-dimensional mesheswhile twisting them, the boundary surface SL is generated between thestator side and the rotor side, and a three-dimensional mesh isgenerated by filling the space between the stator side and the rotorside with polyhedrons including the respective surface elementsconstituting the boundary surface SL, so that a three-dimensional meshwith the stator side rotatable from the boundary surface SL can begenerated for the rotating machine having skew. Moreover, in the casewhere the two-dimensional mesh is composed of quadrangles, sinceportions other than the cylindrical gap G2 are composed of hexahedralelements, the accuracy of calculation performed using thethree-dimensional mesh can be improved compared to the case where allthe portions are composed of tetrahedral elements. Further, since theportion of the cylindrical gap G2 is filled with a plurality ofpolyhedrons periodically in the rotation direction, it is possible togive periodicity to the three-dimensional mesh without requiring aspecial process which is required in the case of using automatic elementdivision.

This embodiment illustrates a method in which two-dimensional meshes aregenerated first, an initial three-dimensional mesh is generated bystacking the generated two-dimensional meshes while twisting them, andthe generated initial three-dimensional mesh is inputted into thethree-dimensional mesh generating apparatus 1 to generate athree-dimensional mesh, but it may be possible to directly generate aninitial three-dimensional mesh and input the initial three-dimensionalmesh into the three-dimensional mesh generating apparatus 1. In thiscase, the inputted initial three-dimensional mesh is divided so that thestator-side mesh surface ST and the rotor-side mesh surface RT aredivided into mutually equal lengths perpendicular to the rotation axisin the direction of the rotation axis, and equally divided into mutuallyequal number of parts in the direction around the rotation axis, andportions other than portions including the stator-side mesh surface STand the rotor-side mesh surface RT may be composed of combinations ofmore irregular polyhedrons, in such a manner that there is a differencein the number or length of the polyhedrons in the direction of therotation axis between the stator side and the rotor side.

Moreover, although this embodiment illustrates the generation of athree-dimensional mesh for a rotating machine with the rotor having atwisted structure with respect to the rotation axis, the presentinvention is not limited to this, and it is also possible to generate athree-dimensional mesh in the same manner for a rotating machine withthe stator having a twisted structure with respect to the rotation axisby using the present invention. Besides, although the rotating machinewith the rotor positioned inside the stator is explained, it is alsopossible to generate a three-dimensional mesh in the same manner for arotating machine with the rotor positioned outside the stator. Further,it is also possible to generate a three-dimensional mesh in the samemanner for a rotating machine having no skew structure by using thepresent invention.

Next, the following description will explain the procedure of a methodfor analyzing the magnetic field of a rotating machine according to thepresent invention. FIG. 16 is a block diagram showing a magnetic fieldanalyzing apparatus for a rotating machine according to the presentinvention. In the figure, 4 represents a magnetic field analyzingapparatus for a rotating machine of the present invention implementedusing a computer, which comprises: a CPU 41 for performing operations; aRAM 42; an external memory device 43 such as a CD-ROM drive; and aninternal memory device 44 such as a hard disk, reads a computer program50 of the present invention from a memory product 5 such as a CD-ROM ofthe present invention by the external memory device 43, stores the readcomputer program 50 into the internal memory device 44, and loads thecomputer program 50 into the RAM 42, and the CPU 41 executes processesnecessary for the magnetic field analyzing apparatus 4 for a rotatingmachine, based on the computer program 50. The magnetic field analyzingapparatus 4 for a rotating machine comprises an input device 45 such asa keyboard or a mouse, and an output device 46 such as a liquid crystaldisplay or a CRT display, and receives operations, such as input ofdata, from an operator.

Moreover, the magnetic field analyzing apparatus 4 for a rotatingmachine comprises a communication interface 47, and may be arranged todownload the computer program 50 of the present invention from a serverdevice 3 connected to the communication interface 47 and execute theprocesses by the CPU 41.

FIG. 17 is a flowchart showing the flow of processes performed by themagnetic field analyzing apparatus 4 for a rotating machine according tothe present invention. The magnetic field analyzing apparatus 4 for arotating machine generates a three-dimensional mesh representing arotating machine to be analyzed by using the above-mentionedthree-dimensional mesh generating method (S100), and rotates the rotorside of the generated three-dimensional mesh from the boundary surfaceSL by one step by shifting the rotor side with respect to the statorside by one element (S200). Next, the magnetic field analyzing apparatus4 connects the rotor side and the stator side at the boundary surface SL(S300), and analyzes the magnetic field of the rotating machine by usinga finite element method (S400). Next, the magnetic field analyzingapparatus 4 decides whether or not to continue the calculation by, forexample, receiving an instruction to finish the calculation from theoperator (S500), and returns to step S200 and further rotates the rotorside by one step if the calculation continues, or finishes the processif the calculation does not continue.

FIG. 18 is a perspective view showing a part of the three-dimensionalmesh rotated. The rotor side is shifted from the boundary surface SL inthe rotation direction by one element with respect to the stator sidefrom the initial state shown in FIG. 18(a), and thus the rotor side isrotated by one step as shown in FIG. 18(b). In the case where the rotorside is further rotated, the rotor side is further shifted by oneelement and thus rotated by one step as shown in FIG. 18(c).

By performing analysis as described above, it is possible to analyze themagnetic field of a rotating machine having skew. Since the analysis isperformed by rotating the rotor side from the boundary surface SL, thecalculation time is shorter compared to a method in which athree-dimensional mesh is generated by an automatic element divisionmethod whenever the rotor side is rotated, and, since the regularthree-dimensional mesh is used, more accurate calculation can beperformed compared to a method using an irregular three-dimensional meshgenerated by the automatic element division method.

In this embodiment, although a mode in which the magnetic fieldanalyzing apparatus 4 for a rotating machine comprises means forgenerating a three-dimensional mesh is illustrated, the presentinvention is not limited to this mode and may be arranged so that themagnetic field analyzing apparatus 4 for a rotating machine does notcomprise means for generating a three-dimensional mesh, receives theinput of a three-dimensional mesh generated by the three-dimensionalmesh generating apparatus 1, and analyzes the magnetic field of therotating machine by using the inputted three-dimensional mesh.

INDUSTRIAL APPLICABILITY

As described above, in the present invention, since portions of thestator side and the rotor side of the three-dimensional mesh which comeinto contact with each other at the boundary surface are composed ofelements having mutually equal size in the rotation direction, athree-dimensional mesh that allows rotation of the rotor side byshifting the elements from the boundary surface can be generated evenfor a rotating machine having skew.

Moreover, the respective elements of the initial three-dimensional meshare hexahedrons, and the calculation accuracy is improved by usinghexahedral elements than by using tetrahedral elements according to thefinite element method, and thus it is possible to generate athree-dimensional mesh with high calculation accuracy.

Further, by associating each of the surface elements constituting thestator-side mesh surface and the rotor-side mesh surface, respectively,with surface elements constituting the boundary surface and filling thespace between the corresponding surface elements with one quadrangularpyramid and four tetrahedrons, it is possible to make thethree-dimensional mesh periodic in the rotation direction withoutrequiring an additional process.

In addition, even for a three-dimensional mesh representing a rotatingmachine having skew, it is possible to generate a three-dimensional meshthat allows rotation of the rotor side by shifting the elements from theboundary surface, has periodicity in the rotation direction, in highcalculation accuracy.

Furthermore, since the magnetic field of the rotating machine isanalyzed by the finite element method using the generatedthree-dimensional mesh, it is possible to perform accurate magneticfield analysis with a shorter calculation time.

1. A method for generating a three-dimensional mesh representing arotating machine with a stator or a rotor having a twisted structure ina direction of a rotation axis of the rotor, including a spatial areabetween the stator and the rotor, by a combination of a plurality ofpolyhedrons, characterized by: generating a two-dimensional mesh inwhich a ring-shaped gap is provided around the rotation axis in thespatial area, portions facing each other with the ring-shaped gaptherebetween are equally divided into mutually equal number of parts,and a stator-side portion and a rotor-side portion, excluding thering-shaped gap, are represented by a combination of a plurality ofpolyhedrons on a plane perpendicular to the rotation axis; generating aninitial three-dimensional mesh by joining together a plurality of thetwo-dimensional meshes with the stator-side portion and the rotor-sideportion relatively rotated on the rotation axis according to the twistedstructure, in the direction of the rotation axis according to a samerule in the stator-side portion and the rotor-side portion; forming aboundary surface constructed by a mesh surface obtained byconcentrically projecting, into a cylindrical gap composed of a stack ofthe ring-shaped gaps, any one of a stator-side mesh surface and arotor-side mesh surface which face each other with the cylindrical gaptherebetween; and filling spaces between the boundary surface and thestator-side mesh surface and rotor-side mesh surface with a plurality ofpolyhedrons, including polyhedrons comprising each of surface elementsconstituting the boundary surface, the stator-side mesh surface and therotor-side mesh surface as one face, to generate the three-dimensionalmesh.
 2. The three-dimensional mesh generating method according to claim1, characterized in that the two-dimensional mesh is composed of acombination of a plurality of quadrangles, and the initialthree-dimensional mesh is generated by joining together a plurality ofthe two-dimensional meshes in the direction of the rotation axis so thatcorresponding nodes in the two-dimensional meshes are connected by astraight line.
 3. The three-dimensional mesh generating method accordingto claim 2, characterized by: dividing each of quadrangular elementsconstituting the boundary surface into two triangular elements arrangedin a direction in which the boundary surface is twisted with respect tothe stator-side mesh surface or the rotor-side mesh surface; connectingeach of nodes constituting the stator-side mesh surface and therotor-side mesh surface to a closest node among a plurality of nodesconstituting the boundary surface by a straight line; and filling aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base.
 4. A method for generating athree-dimensional mesh representing a rotating machine, including aspatial area between a stator and a rotor, by a combination of aplurality of polyhedrons by a computer, characterized by comprisingsteps of: receiving from an input unit and storing into a storage unitan initial three-dimensional mesh in which a cylindrical gap is providedin the spatial area between the stator and the rotor of the rotatingmachine, portions facing each other with the cylindrical gaptherebetween are divided into mutually equal lengths perpendicular to arotation axis of the rotating machine in a direction of the ration axisand equally divided into mutually equal number of parts in a directionaround the rotation axis, and a stator-side portion and a rotor-sideportion of the rotating machine, excluding the cylindrical gap, arerepresented by a combination of a plurality of polyhedrons; forming aboundary surface by a mesh surface obtained by concentricallyprojecting, into the cylindrical gap, any one of a stator-side meshsurface and a rotor-side mesh surface which face each other with thecylindrical gap therebetween, and storing the boundary surface into thestorage unit; dividing each of quadrangular elements constituting theboundary surface into two triangular elements arranged in a directiontilted with respect to each of quadrangular elements constituting thestator-side mesh surface or the rotor-side mesh surface, and storingthem into the storage unit; connecting each of nodes constituting thestator-side mesh surface and the rotor-side mesh surface to a closestnode among a plurality of nodes constituting the boundary surface by astraight line, and storing them into the storage unit; and filling aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base, and storing them into thestorage unit.
 5. A method for analyzing a magnetic field of a rotatingmachine by a finite element method using a three-dimension meshrepresenting the rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons,characterized by generating a three-dimensional mesh representing arotating machine to be analyzed by using the three-dimensional meshgenerating method of any one of claims 1 through 4, rotating a rotorside of the three-dimensional mesh by shifting the elements from theboundary surface, and analyzing the magnetic field by the finite elementmethod.
 6. An apparatus for generating a three-dimensional meshrepresenting a rotating machine, including a spatial area between astator and a rotor, by a combination of a plurality of polyhedrons,characterized by comprising: means for receiving an initialthree-dimensional mesh in which a cylindrical gap is provided in thespatial area between the stator and the rotor of the rotating machine,portions facing each other with the cylindrical gap therebetween aredivided into mutually equal lengths perpendicular to a rotation axis ofthe rotating machine in a direction of the rotation axis and equallydivided into mutually equal number of parts in a direction around therotation axis, and a stator-side portion and a rotor-side portion of therotating machine, excluding the cylindrical gap, are represented by acombination of a plurality of polyhedrons; means for forming a boundarysurface by a mesh surface obtained by concentrically projecting, intothe cylindrical gap, any one of a stator-side mesh surface and arotor-side mesh surface which face each other with the cylindrical gaptherebetween; means for dividing each of quadrangular elementsconstituting the boundary surface into two triangular elements arrangedin a direction in which the boundary surface is tilted with respect tothe stator-side mesh surface or the rotor-side mesh surface; means forconnecting each of nodes constituting the stator-side mesh surface andthe rotor-side mesh surface to a closest node among a plurality of nodesconstituting the boundary surface by a straight line; and means forfilling a space between each of surface elements constituting thestator-side mesh surface and rotor-side mesh surface and a combinationof the two triangular elements connected to the surface element bystraight lines, with four tetrahedrons, including two tetrahedronscomprising each of the two triangular elements as one face, and onequadrangular pyramid comprising the surface element as a base.
 7. Anapparatus for analyzing a magnetic field of a rotating machine by afinite element method using a three-dimensional mesh representing therotating machine, including a spatial area between a stator and a rotor,by a combination of a plurality of polyhedrons, characterized bycomprising: means for generating a three-dimensional mesh representing arotating machine to be analyzed, by using the three-dimensional meshgenerating apparatus of claim 6; and means for rotating the rotor sideof the three-dimensional mesh by shifting the elements from the boundarysurface, and analyzing the magnetic field by the finite element method.8. A computer program for causing a computer to generate athree-dimensional mesh representing a rotating machine, including aspatial area between a stator and a rotor, by a combination of aplurality of polyhedrons, by using an initial three-dimensional mesh inwhich a cylindrical gap is provided in the spatial area between thestator and the rotor of the rotating machine, portions facing each otherwith the cylindrical gap therebetween are divided into mutually equallengths perpendicular to a rotation axis of the rotating machine in adirection of the rotation axis and equally divided into mutually equalnumber of parts in a direction around the rotation axis, and astator-side portion and a rotor-side portion of the rotating machine,excluding the cylindrical gap, are represented by a combination of aplurality of polyhedrons, characterized by comprising steps of: causingthe computer to form a boundary surface by a mesh surface obtained byconcentrically projecting, into the cylindrical gap, any one of astator-side mesh surface and a rotor-side mesh surface which face eachother with the cylindrical gap therebetween; causing the computer todivide each of quadrangular elements constituting the boundary surfaceinto two triangular elements arranged in a direction in which theboundary surface is tilted with respect to the stator-side mesh surfaceor the rotor-side mesh surface; causing the computer to connect each ofnodes constituting the stator-side mesh surface and the rotor-side meshsurface to a closest node among a plurality of nodes constituting theboundary surface by a straight line; and causing the computer to fill aspace between each of surface elements constituting the stator-side meshsurface and rotor-side mesh surface and a combination of the twotriangular elements connected to the surface element by straight lines,with four tetrahedrons, including two tetrahedrons comprising each ofthe two triangular elements as one face, and one quadrangular pyramidcomprising the surface element as a base.
 9. A computer program forcausing a computer to analyze a magnetic field of a rotating machine bya finite element method using a three-dimensional mesh representing arotating machine, including a spatial area between a stator and a rotor,by a combination of a plurality of polyhedrons, characterized bycomprising steps of: causing the computer to generate athree-dimensional mesh representing a rotating machine to be analyzed,by using the computer program of claim 8; and causing the computer torotate a rotor side of the three-dimensional mesh by shifting theelements from the boundary surface and analyze the magnetic field by thefinite element method.
 10. A memory product readable by a computerstoring a computer program for causing a computer to generate athree-dimensional mesh representing a rotating machine, including aspatial area between a stator and a rotor, by a combination of aplurality of polyhedrons, by using an initial three-dimensional mesh inwhich a cylindrical gap is provided in the spatial area between thestator and the rotor of the rotating machine, portions facing each otherwith the cylindrical gap therebetween are divided into mutually equallengths perpendicular to a rotation axis of the rotating machine in adirection of the rotation axis and equally divided into mutually equalnumber of parts in a direction around the rotation axis, and astator-side portion and a rotor-side portion of the rotating machine,excluding the cylindrical gap, are represented by a combination of aplurality of polyhedrons, characterized by storing a computer programcomprising steps of: causing the computer to form a boundary surface bya mesh surface obtained by concentrically projecting, into thecylindrical gap, any one of a stator-side mesh surface and a rotor-sidemesh surface which face each other with the cylindrical gaptherebetween; causing the computer to divide each of quadrangularelements constituting the boundary surface into two triangular elementsarranged in a direction in which the boundary surface is tilted withrespect to the stator-side mesh surface or the rotor-side mesh surface;causing the computer to connect each of nodes constituting thestator-side mesh surface and the rotor-side mesh surface to a closestnode among a plurality of nodes constituting the boundary surface by astraight line; and causing the computer to fill a space between each ofsurface elements constituting the stator-side mesh surface androtor-side mesh surface and a combination of the two triangular elementsconnected to the surface element by straight lines, with fourtetrahedrons, including two tetrahedrons comprising each of the twotriangular elements as one face, and one quadrangular pyramid comprisingthe surface element as a base.
 11. A memory product readable by acomputer storing a computer program for causing a computer to analyze amagnetic field of a rotating machine by a finite element method using athree-dimensional mesh representing a rotating machine, including aspatial area between a stator and a rotor, by a combination of aplurality of polyhedrons, characterized by storing a computer programcomprising steps of: causing the computer to generate athree-dimensional mesh representing a rotating machine to be analyzed,by using the computer program stored in the memory product of claim 10;and causing the computer to rotate a rotor side of the three-dimensionalmesh by shifting the elements from the boundary surface and analyze themagnetic field by the finite element method.